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Re: Natural numbers and vases

Subject	Author
Natural numbers and vases	William
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	FredJeffries
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Mostowski Collapse
Re: Natural numbers and vases	Socratis T.n.p.

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Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sun, 19 Jun 2022 05:17 UTC

On Monday, June 6, 2022 at 8:33:34 PM UTC+2, William wrote:
> On Monday, June 6, 2022 at 3:18:01 PM UTC-3, WM wrote:
> >
> > [I] show that for every number of exchanges, finite or infinite, the O's remain.
> >
> Nope. You only show this for a finite number of exchanges, then claim it holds for "an infinite number of exchanges".

Nonsense. WM is right. Use induction, idiot!

> But the phase

Shut up!

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sun, 19 Jun 2022 05:29 UTC

On Sunday, June 19, 2022 at 6:14:21 AM UTC+2, sergi o wrote:
> On 6/6/2022 3:21 PM, WM wrote:
> >
> > I claim what is obvious: Exchanging X and O will never increase or decrease any of them.

WM is right.

Consider the following sequence of sequences where the n+1-th term results from the n-th term by an exchange of X with an O:

((X, O, O, O, ...), (O, X, O, O, ...), (O, O, X, O, ...), ...)

You really think that there's a term in this sequence without X as a term?

Induction proves that this is not the case.

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Subject: Re: Natural numbers and vases
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Mon, 20 Jun 2022 04:59 UTC

fredag 17 juni 2022 kl. 14:53:51 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Freitag, 17. Juni 2022 um 11:16:34 UTC+2:
> > torsdag 16 juni 2022 kl. 19:53:06 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Mittwoch, 15. Juni 2022 um 07:17:36 UTC+2:
> > > > tisdag 14 juni 2022 kl. 16:25:35 UTC+2 skrev WM:
> > >
> > > > > All FISONs cover all |N. There is no number missing. Hence if all FISONs are subtracted, then there is no number remaining. No union is required. But
> > > > >
> > > > > ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > >
> > > > > Contradiction!
> > > > >
> > > > Wrong, what you want to do is
> > > > |ℕ \ {1} \ {1,2} \ {1,2,3}\ ...|
> > > >
> > > > Which is indeed, 0
> > > Not when only subtracting FISONs smaller than ℕ.
> > >
> > I subtracted all fisons there and it is still 0 :)
> Interesting. Which FISON was the first not leaving ℵo nanumbers?
>
> Regards, WM

Why would there be a first? :) That is an erroneous assumption by you

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Mon, 20 Jun 2022 14:05 UTC

zelos...@gmail.com schrieb am Montag, 20. Juni 2022 um 06:59:31 UTC+2:
> fredag 17 juni 2022 kl. 14:53:51 UTC+2 skrev WM:

> > > I subtracted all fisons there and it is still 0 :)
> > Interesting. Which FISON was the first not leaving ℵo nanumbers?
> >
> Why would there be a first? :) That is an erroneous assumption by you

No, it is mathematics: The endsegments E(k) shrink only by one element per FISON:

∀k ∈ ℕ: E(k+1) = E(k) \ {k} .

Regards, WM

Re: Natural numbers and vases

<5ccfc941-be4b-49bd-84b5-69e650cf5706n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Mon, 20 Jun 2022 14:14 UTC

Fritz Feldhase schrieb am Sonntag, 19. Juni 2022 um 07:30:03 UTC+2:
> On Sunday, June 19, 2022 at 6:14:21 AM UTC+2, sergi o wrote:
> > On 6/6/2022 3:21 PM, WM wrote:
> > >
> > > I claim what is obvious: Exchanging X and O will never increase or decrease any of them.
> WM is right.
>
> Consider the following sequence of sequences where the n+1-th term results from the n-th term by an exchange of X with an O:
>
> ((X, O, O, O, ...), (O, X, O, O, ...), (O, O, X, O, ...), ...)
>
> You really think that there's a term in this sequence without X as a term?
>
> Induction proves that this is not the case.

Or consider the sequence of the first n counted fractions. Do you really think that there is a term in this sequence where all fractions have been counted?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: sergi o - Mon, 20 Jun 2022 15:04 UTC

On 6/20/2022 9:05 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 20. Juni 2022 um 06:59:31 UTC+2:
>> fredag 17 juni 2022 kl. 14:53:51 UTC+2 skrev WM:
>
>>>> I subtracted all fisons there and it is still 0 :)
>>> Interesting. Which FISON was the first not leaving ℵo nanumbers?
>>>
>> Why would there be a first? :) That is an erroneous assumption by you
>
> No, it is mathematics: The endsegments E(k) shrink only by one element per FISON:

wrong. all endsegments are the same size, ℵo .

>
> Regards, WM

Re: Natural numbers and vases

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by: sergi o - Mon, 20 Jun 2022 15:05 UTC

On 6/20/2022 9:14 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 19. Juni 2022 um 07:30:03 UTC+2:
>> On Sunday, June 19, 2022 at 6:14:21 AM UTC+2, sergi o wrote:
>>> On 6/6/2022 3:21 PM, WM wrote:
>>>>
>>>> I claim what is obvious: Exchanging X and O will never increase or decrease any of them.
>> WM is right.
>>
>> Consider the following sequence of sequences where the n+1-th term results from the n-th term by an exchange of X with an O:
>>
>> ((X, O, O, O, ...), (O, X, O, O, ...), (O, O, X, O, ...), ...)
>>
>> You really think that there's a term in this sequence without X as a term?
>>
>> Induction proves that this is not the case.
>
> Or consider the sequence of the first n counted fractions. Do you really think that there is a term in this sequence where all fractions have been counted?
>
> Regards, WM

do you ? who is doing the counting ?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Mon, 20 Jun 2022 15:10 UTC

On Monday, June 20, 2022 at 11:14:40 AM UTC-3, WM wrote:
> consider the sequence of the first n counted fractions. Do you really think that there is a term in this sequence where all fractions have been counted?

There is no such term. So what? The sequence contains every fraction. Consider the sequence of natural numbers. There is no term where all numbers have been counted, The sequence contains every natural. The elements of the set |N are finite. The set |N is infinite.

--
William Hughes

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by: sergi o - Mon, 20 Jun 2022 15:14 UTC

so they have been counted to n, and you stopped.

> Do you really think that there is a term in this sequence where all fractions have been counted?

It is an infinite sequence, what do you think ?

>
> Regards, WM

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by: FromTheRafters - Mon, 20 Jun 2022 15:57 UTC

WM wrote :
> zelos...@gmail.com schrieb am Montag, 20. Juni 2022 um 06:59:31 UTC+2:
>> fredag 17 juni 2022 kl. 14:53:51 UTC+2 skrev WM:
>
>>>> I subtracted all fisons there and it is still 0 :)
>>> Interesting. Which FISON was the first not leaving ℵo nanumbers?
>>>
>> Why would there be a first? :) That is an erroneous assumption by you
>
> No, it is mathematics: The endsegments E(k) shrink only by one element per
> FISON:
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k} .

Endsegments don't shrink.

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Mon, 20 Jun 2022 20:08 UTC

FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:

> Endsegments don't shrink.

They do, compared to their predecessors:
∀k ∈ ℕ: E(k+1) = E(k) \ {k}

Regards, WM

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Mon, 20 Jun 2022 20:14 UTC

William schrieb am Montag, 20. Juni 2022 um 17:10:58 UTC+2:
> On Monday, June 20, 2022 at 11:14:40 AM UTC-3, WM wrote:
> > consider the sequence of the first n counted fractions. Do you really think that there is a term in this sequence where all fractions have been counted?
> There is no such term. So what?

It is claimed that all can be counted by finite indices (not by the sequence).

> The sequence contains every fraction.

Every definable fraction, yes. But not all fractions. Proof:

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
.... ... ... ...

will never gather all fractions in te first column:

1/1, __, __, __, ...
1/2, __, __, __, ...
1/3, __, __, __, ...
2/2, __, __, __, ...
....

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
Date: Mon, 20 Jun 2022 17:10:17 -0400
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by: FromTheRafters - Mon, 20 Jun 2022 21:10 UTC

It happens that WM formulated :
> FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
>
>> Endsegments don't shrink.
>
> They do, compared to their predecessors:
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}

No, they don't. They are all the same size.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Mon, 20 Jun 2022 21:27 UTC

On Monday, June 20, 2022 at 10:08:37 PM UTC+2, WM wrote:
> FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
> >
> > Endsments don't shrik.
> >
> They do
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}

No, they don't: ∀k ∈ ℕ: |E(k+1)| = |E(k)|.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Tue, 21 Jun 2022 02:00 UTC

On Monday, June 20, 2022 at 5:14:12 PM UTC-3, WM wrote:
> ... Proof:
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
> 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> ... ... ... ...
>
> will never gather all fractions in te first column:

So what? If you use a method that cannot get to the bijection, the method will not get to the bijection. However, there are methods that can produce the bijection.

--
William Hughes

Re: Natural numbers and vases

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by: sergi o - Tue, 21 Jun 2022 02:36 UTC

On 6/20/2022 9:00 PM, William wrote:
> On Monday, June 20, 2022 at 5:14:12 PM UTC-3, WM wrote:
>> ... Proof:
>>
>> 1/1, 1/2, 1/3, 1/4, ...
>> 2/1, 2/2, 2/3, 2/4, ...
>> 3/1, 3/2, 3/3, 3/4, ...
>> 4/1, 4/2, 4/3, 4/4, ...
>> 5/1, 5/2, 5/3, 5/4, ...
>> ...
>>
>> 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
>> 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
>> 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
>> 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
>> 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
>> ... ... ... ...
>>
>> will never gather all fractions in te first column:
>
> So what? If you use a method that cannot get to the bijection, the method will not get to the bijection. However, there are methods that can produce the bijection.
>

this is one way and easy, google Cantor's Enumeration of the Rationals

there are other bijections of the fractions on line too in different order

done the Right Way;

http://mathcenter.oxford.emory.edu/site/math125/rationalEnumeration/

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Subject: Re: Natural numbers and vases
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Tue, 21 Jun 2022 05:31 UTC

måndag 20 juni 2022 kl. 16:06:00 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 20. Juni 2022 um 06:59:31 UTC+2:
> > fredag 17 juni 2022 kl. 14:53:51 UTC+2 skrev WM:
>
> > > > I subtracted all fisons there and it is still 0 :)
> > > Interesting. Which FISON was the first not leaving ℵo nanumbers?
> > >
> > Why would there be a first? :) That is an erroneous assumption by you
> No, it is mathematics:

No, your stuff is crankery.

>The endsegments E(k) shrink only by one element per FISON:
>
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k} .
>
> Regards, WM

None of which implies that there is a first one that makes it empty. That is your fautly assumption. That there must be when there isn't

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Tue, 21 Jun 2022 11:50 UTC

FromTheRafters schrieb am Montag, 20. Juni 2022 um 23:10:30 UTC+2:
> It happens that WM formulated :
> > FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
> >
> >> Endsegments don't shrink.
> >
> > They do, compared to their predecessors:
> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> No, they don't. They are all the same size.

Nevertheless they do
∀k ∈ ℕ: E(k+1) = E(k) \ {k}
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Tue, 21 Jun 2022 11:50 UTC

Fritz Feldhase schrieb am Montag, 20. Juni 2022 um 23:27:26 UTC+2:
> On Monday, June 20, 2022 at 10:08:37 PM UTC+2, WM wrote:
> > FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
> > >
> > > Endsments don't shrik.
> > >
> > They do
> >
> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> No, they don't: ∀k ∈ ℕ: |E(k+1)| = |E(k)|.

They do
∀k ∈ ℕ: E(k+1) = E(k) \ {k}
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Tue, 21 Jun 2022 11:53 UTC

William schrieb am Dienstag, 21. Juni 2022 um 04:00:33 UTC+2:
> On Monday, June 20, 2022 at 5:14:12 PM UTC-3, WM wrote:
> > ... Proof:
> >
> > 1/1, 1/2, 1/3, 1/4, ...
> > 2/1, 2/2, 2/3, 2/4, ...
> > 3/1, 3/2, 3/3, 3/4, ...
> > 4/1, 4/2, 4/3, 4/4, ...
> > 5/1, 5/2, 5/3, 5/4, ...
> > ...
> >
> > 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
> > 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
> > 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
> > 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
> > 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> > ... ... ... ...
> >
> > will never gather all fractions in te first column:
> So what? If you use a method that cannot get to the bijection, the method will not get to the bijection.

But I use a method that is "proved" to be a bijection.

> However, there are methods that can produce the bijection.

Above you see is the most famous one, designed by Cantor.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 21 Jun 2022 12:13 UTC

On Tuesday, June 21, 2022 at 1:51:01 PM UTC+2, WM wrote:
> FromTheRafters schrieb am Montag, 20. Juni 2022 um 23:10:30 UTC+2:
> > It happens that WM formulated :
> > > FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
> > >
> > >> Endsegments don't shrink.
> > >
> > > They do, compared to their predecessors:
> > > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > >
> > No, they don't. They are all the same size.
> >
> Nevertheless they do

No, they don't.

> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

So what?

Since k ∈ ℕ: |E(k)| = |E(k)| - 1 (=aleph_0) we have

∀k ∈ ℕ: |E(k+1)| = |E(k)|.

What's the matter with you, you psychotic asshole?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 21 Jun 2022 12:14 UTC

On Tuesday, June 21, 2022 at 1:51:05 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Montag, 20. Juni 2022 um 23:27:26 UTC+2:
> > On Monday, June 20, 2022 at 10:08:37 PM UTC+2, WM wrote:
> > > FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
> > > >
> > > > Endsegments don't shrink.
> > > >
> > > They do
> > >
> > > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > >
> > No, they don't: ∀k ∈ ℕ: |E(k+1)| = |E(k)|.
> >
> They do

No, they don't.

> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

So what?

Since k ∈ ℕ: |E(k)| = |E(k)| - 1 (=aleph_0) we have

∀k ∈ ℕ: |E(k+1)| = |E(k)|.

What's the matter with you, you psychotic asshole?

Re: Natural numbers and vases

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by: FromTheRafters - Tue, 21 Jun 2022 13:17 UTC

WM formulated on Tuesday :
> FromTheRafters schrieb am Montag, 20. Juni 2022 um 23:10:30 UTC+2:
>> It happens that WM formulated :
>>> FromTheRafters schrieb am Montag, 20. Juni 2022 um 17:57:15 UTC+2:
>>>
>>>> Endsegments don't shrink.
>>>
>>> They do, compared to their predecessors:
>>> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
>> No, they don't. They are all the same size.
>
> Nevertheless they do
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> ∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

Use Cardinal Arithmetic when working with infinite cardinalities.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Tue, 21 Jun 2022 13:38 UTC

On Tuesday, June 21, 2022 at 8:53:20 AM UTC-3, WM wrote:
> William schrieb am Dienstag, 21. Juni 2022 um 04:00:33 UTC+2:
> > On Monday, June 20, 2022 at 5:14:12 PM UTC-3, WM wrote:
> > > ... Proof:
> > >
> > > 1/1, 1/2, 1/3, 1/4, ...
> > > 2/1, 2/2, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ...
> > > 5/1, 5/2, 5/3, 5/4, ...
> > > ...
> > >
> > > 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
> > > 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
> > > 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> > > ... ... ... ...
> > >
> > > will never gather all fractions in te first column:
> > So what? If you use a method that cannot get to the bijection, the method will not get to the bijection.
> But I use a method that is "proved" to be a bijection.

Nope. The method you use does not get to the bijection designed by Cantor. Every step only produces part of the bijection. You have not defined a state where all steps are done.

--
William Hughes

Re: Natural numbers and vases

<t8sig4$2f6$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 21 Jun 2022 09:53:04 -0400
Organization: Peripheral Visions
Lines: 41
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References: <06d0f903-3235-4be2-8653-85563102704fn@googlegroups.com> <c952a918-bf4f-4c9c-8c6c-34bb98452758n@googlegroups.com> <858b03d6-fd8e-45f0-967e-8663b5d6cc47n@googlegroups.com> <94779ba4-ee8c-43eb-86c8-44b2a253d2e2n@googlegroups.com> <8d2232cc-ad35-460d-90da-80c0deef6847n@googlegroups.com> <fd833c8d-922e-468f-881c-a33e721e22a2n@googlegroups.com> <a17c74e3-e3a9-4e5d-88a5-8d4f460a2c32n@googlegroups.com> <f1d6be67-cfc2-4808-882f-fbd23ecde1b1n@googlegroups.com> <849a4540-610e-40c3-90e8-8a1e69ce276en@googlegroups.com> <t8m7qc$dba$1@gioia.aioe.org> <f18ec10d-8f6d-4b64-a949-926f6479e2ddn@googlegroups.com> <5ccfc941-be4b-49bd-84b5-69e650cf5706n@googlegroups.com> <3061b621-c848-4fe5-accc-457f5596b4ebn@googlegroups.com> <8bbc710e-cefa-451e-a57d-9822934a31c4n@googlegroups.com> <c11a3e07-2047-467b-aefa-94b77714f0f9n@googlegroups.com> <a5d45cf4-305f-4780-aff5-9ef65dd7c1ben@googlegroups.com> <d85ec2bc-9ab1-484f-a444-2868b275b9bbn@googlegroups.com>
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by: FromTheRafters - Tue, 21 Jun 2022 13:53 UTC

William explained on 6/21/2022 :
> On Tuesday, June 21, 2022 at 8:53:20 AM UTC-3, WM wrote:
>> William schrieb am Dienstag, 21. Juni 2022 um 04:00:33 UTC+2:
>>> On Monday, June 20, 2022 at 5:14:12 PM UTC-3, WM wrote:
>>>> ... Proof:
>>>>
>>>> 1/1, 1/2, 1/3, 1/4, ...
>>>> 2/1, 2/2, 2/3, 2/4, ...
>>>> 3/1, 3/2, 3/3, 3/4, ...
>>>> 4/1, 4/2, 4/3, 4/4, ...
>>>> 5/1, 5/2, 5/3, 5/4, ...
>>>> ...
>>>>
>>>> 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
>>>> 1/1, 3/1, 4/1, 1/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ...
>>>> 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ... 3/1, 3/2, 3/3, 3/4, ...
>>>> 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
>>>> 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
>>>> 1/3, 4/2, 4/3, 4/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ...
>>>> 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ... ... ... ... ...
>>>>
>>>> will never gather all fractions in te first column:
>>> So what? If you use a method that cannot get to the bijection, the method
>>> will not get to the bijection.
>> But I use a method that is "proved" to be a bijection.
>
> Nope. The method you use does not get to the bijection designed by Cantor.
> Every step only produces part of the bijection. You have not defined a state
> where all steps are done.

He doesn't want to because he wants to shoehorn his dark numbers in
there. He assumes each natural number has infinitely many successors
when axiomatically he is only guaranteed one successor for each natural
number.

He's happy to assume induction in the case of each natural number
having infinitely many (Aleph_zero) successors but denies induction
when we want to frame the notion of *all* as opposed to *each and
every* describing the set as opposed to its elements.

It might just be that he is incapable of grasping these concepts.

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